The calculus controversy was an argument between 17th-century
mathematicians Isaac Newton and Gottfried Leibniz (begun or fomented
in part by their disciples and associates – see Development of the
quarrel below) over who had first invented calculus. It is a question
that had been the cause of a major intellectual controversy over who
first discovered calculus, one that began simmering in 1699 and broke
out in full force in 1711.

I'm just curious if in the field of mathematics it means one thing to invent and another to discover or if they go totally hand in hand.

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4

I don't think there is a strict distinction in everyday language, and there isn't one in mathematics either. I'd say an invention is just a certain form of discovery, namely if you discover that doing this-or-that will have such-and-such desirable consequences, then your discovery is an invention. In math, the desirable consequences could be that you can now solve or understand something or other more easily. (Patent law has its own definition of "invention", incomprehensible to man).
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Henning MakholmNov 13 '11 at 2:53

2

@simplicity I believe this to be a relevant question. Understanding the nature of mathematics is critical. That being said, this question could fit in both philosophy and mathematics, but I believe it fits best here
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analysisjNov 13 '11 at 3:20

2

@simplicity This is not debate. It is a conversation wherein someone is trying to understand an idea pertaining to mathematics. I would appreciate constructive comments, not insults, especially when they are directed at the original asker.
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analysisjNov 13 '11 at 3:35

3

It's a philosophical question rather than a mathematical one. But I'm not convinced that we should regard it (as "simplicity" suggests) as "stupid".
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Michael HardyNov 13 '11 at 4:27

I just want to point to the fact that this is indicative of a somewhat bigger question. Is mathematics simply descriptive of reality or does it exist on its own in a Platonic existence? For instance, was Fermat's Last Theorem true before it was proved by Wiles? Mario Livio wrote an interesting book exploring this question. It is called Is God a Mathematician. He concludes that certain concepts may be invented, such as calculus, but then the results are discovered as inexorable deductions from the invention.

It's not inconceivable that it is possible to rigorously define the concepts discover and invent without entirely loosing what is tried to capture with the intuitive idea. Any platonist would agree that the structure of the integers, say locations of the prime numbers are discovered. However the integers have many isomorphic representations, say set-theoretic and peano axiomatic. One can argue that these representations are invented by man, but the background structure that governs these representations are discovered.

Groups are a good example, they have many isomorphic representations, but the same structure.